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On the Symmetric Homology of Algebras

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Degree
Doctor of Philosophy, Ohio State University, Mathematics, .
Abstract
The theory of symmetric homology, in which the symmetric groups ${\sum }_{k}^{op}$, for k ≥ 0, play k the role that the cyclic groups do in cyclic homology, begins with the definition of the category ΔS, containing the simplicial category Δ as subcategory. Symmetric homology of a unital algebra, A, over a commutative ground ring, k, is defined using derived functors and the symmetric bar construction of Fiedorowicz. If A = k[G] is a group ring, then HS*(k[G]) is related to stable homotopy theory. Two chain complexes that compute HSâˆ—(A) are constructed, both making use of a symmetric monoidal category ΔS+ containing ΔS, which also permits homology operations to be defined on HS*(A). Two spectral sequences are found that aid in computing symmetric homology. In the second spectral sequence, the complex ${\mathit{Sym}}_{*}^{\left(p\right)}$ is constructed. This complex turns out to be isomorphic to the suspension of the cycle-free chessboard complex, Ωp+1, of Vrećica and Źivaljević. Recent results on the connectivity of Ωn imply finite-dimensionality of the symmetric homology groups of finite- dimensional algebras. Finally, an explicit partial resolution is presented, permitting the calculation of HS0(A) and HS1(A) for a finite-dimensional algebra A.
Subject Headings
Mathematics
Keywords
Symmetric Homology; Functor Homology; Crossed Simplicial Groups; Chessboard Complexes; Homology Operations; GAP
Committee / Advisors
Zbigniew Fiedorowicz (Advisor)
Dan Burghelea (Committee Member)
Roy Joshua (Committee Member)
Avi Benatar (Committee Member)
Pages
166p.

Document number: osu1218237992
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